Solving $x^2-10x-11=0$ A Step-by-Step Guide

Hey guys! Today, we're diving deep into the fascinating world of quadratic equations, and we're going to tackle the equation $x^2 - 10x - 11 = 0$. We'll explore different methods to solve it and, most importantly, connect it to the given options. Trust me, by the end of this article, you’ll be a quadratic equation whiz!

Understanding Quadratic Equations

Before we jump into solving this specific equation, let's take a step back and understand what a quadratic equation really is. In simple terms, a quadratic equation is a polynomial equation of the second degree. What does that mean? It means the highest power of the variable (usually 'x') is 2. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where a, b, and c are constants (numbers), and 'a' is not equal to zero (otherwise, it would be a linear equation). Now, in our case, we have $x^2 - 10x - 11 = 0$. Notice that 'a' is 1, 'b' is -10, and 'c' is -11. Recognizing this form is the first step in conquering these equations.

So, why are quadratic equations important? Well, they show up everywhere in the real world! From physics (projectile motion) to engineering (designing bridges) to even finance (modeling growth), quadratic equations are essential tools. Learning how to solve them opens up a whole new world of problem-solving abilities. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. We'll focus on the method that aligns best with the given options – completing the square – but it's always good to know your options! Understanding these equations is not just about math; it's about understanding how the world works around us. The versatility of quadratic equations makes them a cornerstone of mathematical education and practical application. Whether you're calculating the trajectory of a ball or optimizing a business model, the principles behind solving quadratic equations remain consistent and valuable. So, let's get our hands dirty and solve this equation together!

Method 1: Completing the Square

Now, let's get to the heart of the matter – solving $x^2 - 10x - 11 = 0$ by completing the square. This method is super useful because it transforms the quadratic equation into a form that's easy to solve. The idea behind completing the square is to manipulate the equation so that one side becomes a perfect square trinomial. What's a perfect square trinomial, you ask? It's a trinomial (an expression with three terms) that can be factored into the square of a binomial. For example, $x^2 + 2x + 1$ is a perfect square trinomial because it can be factored into $(x + 1)^2$.

Let's walk through the steps. First, we'll move the constant term (-11) to the right side of the equation:

$x^2 - 10x = 11$

Next, we need to figure out what to add to both sides to complete the square on the left side. Here's the trick: take half of the coefficient of the x term (which is -10), square it, and add it to both sides. Half of -10 is -5, and (-5)^2 is 25. So, we add 25 to both sides:

$x^2 - 10x + 25 = 11 + 25$

Now, the left side is a perfect square trinomial! It can be factored into $(x - 5)^2$. The right side simplifies to 36:

$(x - 5)^2 = 36$

Boom! We've completed the square. Notice how this form matches one of the options given in the problem. We're one step closer to cracking this!

Completing the square might seem a bit magical at first, but it's a powerful technique rooted in algebraic principles. By strategically adding a constant to both sides of the equation, we transform a complex expression into a simple, solvable form. This method not only helps in solving quadratic equations but also provides a deeper understanding of their structure. It's like turning a tangled mess into a neat, organized package. And the best part? Once you've mastered completing the square, you'll see its applications in various other mathematical contexts, reinforcing its value as a fundamental problem-solving tool. So, keep practicing, and you'll be completing squares like a pro in no time!

Matching with the Options

Alright, we've successfully transformed the original equation $x^2 - 10x - 11 = 0$ into $(x - 5)^2 = 36$ by completing the square. Now, let's circle back to the options provided in the problem and see which one matches our result. This is where our hard work pays off!

The options were:

A. $(x - 10)^2 = 21$ B. $(x - 5)^2 = 36$ C. $(x - 5)^2 = 21$ D. $(x - 10)^2 = 36$

Take a close look. Which one is an exact match? Option B, right? $(x - 5)^2 = 36$ is precisely what we got after completing the square. That means option B is the correct answer!

You see, sometimes math problems aren't just about finding the final solution (the value of x in this case). They're also about understanding the process and recognizing equivalent forms of the equation. In this problem, the key was to manipulate the equation into a specific format, and then identify that format among the choices. This skill is invaluable because it tests your conceptual understanding rather than just your calculation ability. It's like being able to speak different dialects of the same mathematical language. The more dialects you understand, the better you can communicate and solve problems!

So, give yourself a pat on the back if you followed along and understood how we arrived at option B. You're becoming a master of quadratic equations!

Solving for x (Optional, but Recommended!)

While we've already identified the correct option, let's go the extra mile and actually solve for x in the equation $(x - 5)^2 = 36$. This will give us a deeper understanding of the equation and reinforce our problem-solving skills. Plus, it's just plain fun to see the final answers!

To solve for x, we need to